An Introduction to Field Quantization - download pdf or read online

By Y. Takahashi

ISBN-10: 0080128246

ISBN-13: 9780080128245

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71) is practically zero, then the electric field strength can be measured exactly. In the region of a long wavelength where the energy of the individual photons is sufficiently small, the number of photons contained in the field with a finite energy becomes very large, so that the classical picture is recovered. The essential difference between the quantized and the classical fields can be recognized also in the problem of phase determination. 74) that is to say, the number of photons and the phase of wave cannot be measured simultaneously accurately.

This means that whenever a or κ occurs twice in any term a summation over these indices is to be taken. e. A%\x-x') = A%y'(x-x'), A(-::\X-X') = A(-::;\X-X·). 5) A homogeneous displacement of all atoms in any given direction causes no change so that ΣΑΖΡ(Χ-Χ') = 0. 6) J C' At this stage it is convenient to introduce the reciprocal lattice vectors (1, j , k cyclic). 8) with /i, = 0, ± 1, . . 9) are represented by kn = n1b1jN1 + n2bz/N2+n3bz/N3. 10) Let us now introduce the quantity F{::>\k) = Σ A%\x x' - x') exp ik(x - jcf).

54) ηρ[-β(Η-(ρ\Η\ο))] and /5 _ 1 = kT (k = the Boltzmann constant ; T = the absolute 28 Non-relativistic Fields 2] temperature), and Tr stands for the trace. 55) has eigenvalues 0, 1,2, . . 58) where the factor 2 comes from the two directions of polarization. Hence, the radiation energy density is H__ 2 r dkœ(k)/(eMV-l). 60) Thus we are able to understand the duality of wave and particle. It should be stressed, however, that the quantized radiation field differs essentially from the classical electromagnetic field in its behaviour, as will be seen below.

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An Introduction to Field Quantization by Y. Takahashi


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